radius and for the virial mass. Using this convention, the mean density is given by Other conventions for the overdensity constant include , or , depending on the type of analysis being done, in which case the virial radius and virial mass is signified by the relevant subscript. Defining the virial mass Given the virial radius and the overdensity convention, the virial mass can be found through the relation If the convention that is used, then this becomeswhere is the Hubble parameter as described above, and G is the gravitational constant. This defines the virial mass of an astrophysical system. Applications to dark matter halos Given and , properties of dark matter halos can be defined, including circular velocity, the density profile, and total mass. and are directly related to the Navarro–Frenk–White (NFW) profile, a density profile that describes dark matter halos modeled with the cold dark matter paradigm. The NFW profile is given bywhere is the critical density, and the overdensity (not to be confused with ) and the scale radius are unique to each halo, and the concentration parameter is given by . In place of , is often used, where is a parameter unique to each halo. The total mass of the dark matter halo can then be computed by integrating over the volume of the density out to the virial radius : From the definition of the circular velocity, we can find the circular velocity at the virial radius :Then the circular velocity for the dark matter halo is given bywhere . Although the NFW profile is commonly used, other profiles like the Einasto profile and profiles that take into account the adiabatic contraction of the dark matter due to the baryonic content are also used to characterize dark matter halos. To compute the total mass of the system, including stars, gas, and dark matter, the Jeans equations need to be used with density profiles for each component. See also Dark matter halo Jeans equations Navarro–Frenk–White profile Virial theorem References Stellar astronomy Galactic astronomy Extragalactic astronomy